Practical Stabilization of Driftless Systems on Lie Groups: The Transverse Function Approach

Transcription

1 1496 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003 Practical Stabilization of Driftless Systems on Lie Groups: The Transverse Function Approach Pascal Morin and Claude Samson Abstract A general control design approach for the stabilization of controllable driftless nonlinear systems on finite dimensional Lie groups is presented The approach is based on the concept of bounded transverse functions, the existence of which is equivalent to the system s controllability Its outcome is the practical stabilization of any trajectory, ie, not necessarily a solution of the control system, in the state space The possibility of applying the approach to an arbitrary controllable smooth driftless system follows in turn from the fact that any controllable homogeneous approximation of this system can be lifted (via a dynamic extension) to a system on a Lie group Illustrative examples are given Index Terms Feedback law, Lie groups, nonlinear systems, stabilization I INTRODUCTION L ET (1) denote a control system on a finite-dimensional connected Lie group, left-invariant smooth vector fields (vf) which satisfy the Lie algebra rank condition (LARC) so that the system is controllable when The drift term is viewed here as a known or measured perturbation belonging to the tangent space of at We assume only the continuity of the function Obviously, the main case of interest is when when, for instance The problem addressed in this paper is the practical stabilization of the identity element via the asymptotic stabilization of a compact set contained in an arbitrary small neighborhood of Prior to commenting upon this particular control objective, it may be useful to explain why the Lie group framework is chosen here A common motivation is that various physical systems are naturally modeled as systems invariant on Lie groups Rigid bodies in space and cart-like vehicles are well-known examples Invariance on a Lie group is a strong geometrical and structural property which is understandably useful for control design purposes A second motivation is related to the possibility of locally approximating any smooth controllable driftless system Manuscript received July 15, 2002; revised February 18, 2003 Recommended by Associate Editor W Kang The authors are the Institut National de Recherche en Automatique et Informatique (INRIA), Sophia-Antipolis Cedex, France ( PascalMorin@inriafr; ClaudeSamson@inriafr) Digital Object Identifier /TAC by a controllable homogeneous driftless system which can be lifted, via a suitable dynamic extension, to a system invariant on a Lie group This will be explained in more details further in the paper After having recalled the generality of systems on Lie groups, our next and final argument is that the Lie group framework is particularly well adapted to the development and exposition of the transverse function (tf) control approach presented here The extensive use of the specific properties associated systems on Lie groups in the proofs of the main results reported in the present paper is, by itself, a good illustration of this Let us now focus on control issues and put the practical stabilization objective in perspective the research effort devoted to driftless controllable systems While controllability properties of these systems have been known for a long time as a consequence of the classical Chow theorem [9] algorithms to compute open-loop controls in order to steer the system from one point to another have been proposed more recently various approaches: use of highly oscillatory control inputs [23], [35], [44], explicit calculation of steering trajectories in the case of nilpotent systems [20] and differentially flat systems [13], [27], particularization of these methods to systems on Lie groups [21], possibly a drift vf [6] This paper does not focus on issues related to open-loop control techniques, such as path planning, or the characterization/construction of optimal paths between two points, or the determination of algorithms which compute feasible approximations of a finite-length arbitrary curve Although the proposed control approach may be useful to address some of these problems in a novel way (the path approximation problem, to cite one of them), its primary application is feedback stabilization The problem of asymptotic stabilization via state feedback control of an equilibrium point of (1), when, has also attracted much attention during the last decade Many of the studies on the subject have found a challenge and a motivation in Brockett s theorem [5] according to which, if and the control vf evaluated at are linearly independent, no smooth or even continuous pure state feedback can make this equilibrium point asymptotically stable Different types of feedback laws have been considered to circumvent the difficulty although not all of them guarantee Lyapounov stability Discontinuous feedbacks [1], [3], [8], [22] and hybrid feedbacks [2], [30], [41] are two possibilities Another one, more related to the present approach, consists of using continuous time-varying feedbacks [39], [10], [36], [45], [40], [28], [37], [31], [29] An early survey on the control of nonholonomic systems, whose kinematic models are nonlinear driftless systems, can also be found in [4] /03$ IEEE

2 MORIN AND SAMSON: PRACTICAL STABILIZATION OF DRIFTLESS SYSTEMS ON LIE GROUPS 1497 The abundance of literature on the subject may convey the feeling that the problems are, by now, well understood and have received satisfactory solutions We believe on the contrary that some important issues have been left in the shade One of these issues concerns the compromise between speed of convergence and robustness of the stability property against modeling errors For a few driftless systems, this type of robustness can be ensured by using a Lipschitz continuous time-varying feedback law [26], but this implies slow not exponential convergence to the origin for most of the system s trajectories On the other hand, to our knowledge and understanding, no robust exponential stabilizer has been proposed until now It is in fact possible that such a feedback does not exist for systems which do not satisfy Brockett s condition A result in this direction has been proved in [25] Efforts to circumvent the difficulty, by considering hybrid continuous/discrete time feedback laws [2], [30], have only brought partial results For instance, such feedbacks can be made robust to unmodeled dynamics, but stability of the desired equilibrium is not robust against discretization uncertainties Another issue is related to the trajectory stabilization problem This problem is usually easier than point stabilization in particular, the linearized approximation of the associated error system may be controllable and various feedback solutions have been proposed for specific classes of nonlinear driftless systems, especially in the robotics literature [17], [40], [7] Asymptotic stabilization is usually obtained under some conditions upon the reference trajectory Typically, it should not converge to a fixed point One could have hoped for the existence of a feedback law which would have uniformly guaranteed asymptotic stability independently of the considered trajectory just as for linear controllable systems but negative results concerning this existence issue have been proven [24] This has clear consequences in mobile robotics, when the control objective is the tracking of a reference vehicle whose trajectory is not known in advance The aforementionned difficulties suggest to us that, for nonlinear driftless systems, the classical objective of asymptotic stabilization, which underlies a large part of feedback stabilization theory, is at the same time too restrictive and too constraining The guideline followed in this paper is that practical stabilization can be a more realistic control objective to pursue, and also a touchstone principle for the definition of new approaches for the stabilization of driftless systems Being less constraining than the one of asymptotic stabilization, it encompasses many control solutions previously proposed in the literature and leaves the door open to other solutions The approach here considered for practical stabilization is based on the existence of bounded functions which are transverse to a set of vf [34] Intuitively, one can make a comparison between this approach and the general open-loop control design algorithm developed in [23], [44] In those papers, the idea was to add virtual control inputs involving sinusoids fixed and high enough frequencies in order to ensure uniform boundedness of tracking errors by a prespecified threshold Here, the threshold is directly related to the size of periodic transverse functions whose associated frequencies are the new inputs In comparison the approach developed in [23] and [44], we would like to stress that: 1) our approach yields feedback laws; 2) ultimate boundedness of the tracking errors by a prespecified threshold, uniformly wrt the reference trajectory, is guaranteed; and 3) the control frequencies may tend to zero, even when the reference trajectory is not a solution of the control system, so that oscillations are not systematic this will be illustrated by simulations results at the end of this paper Let us also mention that the idea of frequency adaptation which underlies the feedback control solutions proposed here can be traced back, in the context of mobile robots, to [12], a work itself adapted from control techniques used for induction motors [11] Concerning the concept of transversality used throughout this paper, the basic result of equivalence between the existence of a bounded function whose partial derivatives are transversal to a set of smooth vf and the satisfaction by these vf of the LARC was first proved, to our knowledge, in [34] However, the way of using this result for control purposes is little developed in that reference, and the importance of the Lie group framework in order to properly develop the tf control approach was not identified at that time For the sake of clarity and precision, a discussion about the contribution of the present paper respect to [34] is postponed to Remark 5 in Section VI, after the exposition of the main results The paper is organized as follows The tf control approach is first illustrated on a simple example in Section II In the next section, notations used thereafter are specified and a few definitions about invariant systems on Lie groups are recalled The main technical result about the existence and construction of transverse functions is stated in Theorem 1 in Section IV Then, in Section V, the concept of transverse function is used to solve the practical stabilization problem evoked at the beginning of this introduction In Section VI, we show how the approach applies to systems homogeneous vf, after precising in a proposition how any homogeneous system can be lifted to a system invariant on a Lie group Theorem 1 combined this proposition contains the main result in [34] according to which controllability is equivalent to the existence of transverse functions Finally, the approach is illustrated by examples in Section VII II SIMPLE EXAMPLE Consider the perturbed three-dimensional chained system two inputs and state vector It is known, and simple to verify, that the unperturbed chained system ( ) is controllable Consider the following function which associates an element of the torus a point in The Euclidean norm of is clearly uniformly bounded by a number commensurable, and it uniformly tends to zero when tends to zero This function is of particular interest to us because (2) (3)

3 1498 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003, so that, whenever, the vectors, and form a basis of, for every In other words, the gradient of is transversal to the directions given by et when these vf are evaluated at Note that what is remarkable is not so much the property of transversality by itself the unbounded function also trivially satisfies this property but the combination of transversality and boundedeness of the function In order to explain the usefulness of such a function for the control of (2), we introduce the following vector: One easily verifies that, along the solutions of (2) and Using the fact that both matrices and are invertible for every and, and viewing as an extra control input variable, the preliminary feedback denoting a free vector in, transforms (5) into the simple decoupled linear system Asymptotic stabilization of the origin of the previous system obviously poses no difficulty One can take, for example, to ensure exponential stabilization of In view of (4), this control yields exponential convergence of to zero and, furthermore, asymptotic stabilization of for (2) if, for insatnce, Itis in this sense that the practical stabilization of is achieved for this system Since is almost equal to when is small, a possible interpretation of the above control solution is that it nearly globally linearizes the initial nonlinear system two control inputs into the trivial linear system (quoted words are important here) Following this interpretation, the time derivative of provides the extra control input which allows instantaneous steering in any direction of the state space Since is an element of the torus, its time derivative may also be viewed as a frequency variable At this point, the specific expression (4) of, the dynamics of which can be globally linearized and decoupled contrary to the apparently simpler choice, may seem a little mysterious In this respect, the generalization of the approach will reveal the central role played by the Lie group invariance property of the chained system For instance, we will see that (4) (4) (5) (6) (7) may equally be written as, denoting the group operation on respect to which the chained system is invariant, and the inverse of respect to this operation III RECALLS AND NOTATION Let us introduce some notation used hereafter is the Euclidean norm in Let denote a function of the variables and, we write (resp, )if as (resp if in some neighborhood of ) uniformly respect to belonging to compact sets The tangent space of a manifold at a point is denoted as The Lie algebra generated by vf is denoted as The differential of a smooth mapping at a point is denoted as is the torus of dimension, We shall also use standard notation relative to Lie groups; see, eg, [15] for more details on this topic Recall that a Lie group is a differentiable manifold endowed a smooth group operation denotes a finite-dimensional connected Lie group, Lie algebra of left-invariant vf As usual, for the sake of lightening the notation, and unless specifically indicated otherwise, the group product of two elements and of will be denoted by The identity element of is denoted by, ie, The inverse of is denoted by, ie, Left and right translations are denoted by and respectively, ie, If is the solution at time of initial condition The adjoint representation of is Ad, ie, for defined by By extension, we define The differential of Ad is ad, and, the Lie bracket of and Recall that a vf on a Lie group is left-invariant iff ; a control system on is said to be left-invariant if the control vf s are left-invariant; if and are two solutions of a left-invariant system obtained by applying the same control, then For a proper general exposition of the tf control approach, the following definition of a graded basis of is also needed Definition 1: Let denote independent vf such that Define inductively

4 MORIN AND SAMSON: PRACTICAL STABILIZATION OF DRIFTLESS SYSTEMS ON LIE GROUPS 1499, and let Agraded basis of associated is an ordered basis of associated two mappings such that the following hold 1) For any 2) For and, and With any graded basis of, one can associate a weight vector defined by It follows from Definition 1 that and, IV TRANSVERSE FUNCTIONS The main result about the existence and construction of transverse functions is the following Theorem 1: Let denote a Lie group of dimension Lie algebra Let denote independent vf Then, the following properties are equivalent 1) 2) For any neighborhood of in, there exists a function such that, for any The authors know from their experience in robotics that this extra degree of freedom is useful in practice Another possibility, also important in practice because it can significantly reduce the calculation complexity, consists in using, in either (10) or (12), vf and associated a suitable homogeneous approximation of the control system This is illustrated by the example of a unicycle in Section VII The choice of the parameters is further specified in Lemma 3 used in the proof of the theorem Proof of Theorem 1: For the proof of (Property 2 Property 1), we refer the reader to [34] This is a direct consequence of the Frobenius Theorem To prove that Property 1 implies Property 2, we only have to show that (8) is satisfied the function defined by (9) (10), for some We indicate later the values of the parameters main steps of the proof in the form of three lemmas By standard calculations, we first prove the following Lemma 1: There exist analytic functions such that (13) (14) Furthermore, denoting a graded basis of, a possible choice for is given by (8) From this lemma, we then prove the following Lemma 2: There exist analytic functions such that (15) (9) defined by (10) for some positive real numbers As in [34], functions which satisfy (8) are called transverse to the vf, or just transverse functions when no ambiguity is possible Theorem 1 is coordinate-free When providing a system of coordinates in, relation (8) means that the square matrix (11) is invertible for every There are many ways to derive transverse functions; see, for instance, [33] for an alternative expression of such a function Also, in (10), one can introduce two parameters and, instead of the single parameter This yields (12) (16) where if, and Note that, if all O and o terms in the previous expressions were equal to zero, then Theorem 1 would follow directly from (15) (16) and from the fact that is a basis of Although this is not the case, it is not very difficult to show that these terms can be neglected provided that the s are adequately chosen, as stated in the following lemma Lemma 3: There exist numbers, and, such that choosing, yields (17) In view of (15), (17) is equivalent to (8), so that the proof of the theorem follows The proofs of the previous lemmas are reported in Appendix A

5 1500 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003 Note that Lemma 3 does not state that any set of small enough numbers is suitable, nor does it imply that these numbers have to be small For instance, if the vf are homogeneous wrt some dilation, the lemma holds for any V APPLICATION TO PRACTICAL STABILIZATION A Point Stabilization Consider (1) and assume out loss of generality that are independent We show in the following proposition how the concept of transverse functions can be used to design control laws for System (1) that make practically stable Let us remark that no assumption is made on For instance, when the projection of onto does not tend to zero as tends to infinity, no control law can make an equilibrium of the system, and asymptotic stabilization of this element is pointless Proposition 1: Let denote a transverse function Then along any solution of (1), and along any trajectory in (18) Moreover, if is a vf on, then the dynamic feedback law defined by yields the satisfaction of the following equation: (19) (20) along any solution of the controlled system Proof: The proof of (18) is easily obtained by differentiating the equality and using the identities By applying the feedback law (19) to (1), one deduces (20) from (18) An immediate corollary of the aforementioned proposition is that it suffices to choose for any vf which asymptotically stabilizes to ensure asymptotic stability of this point for the system (20) and, therefore, convergence of to for (1) Furthermore, asymptotic stability of for (1) controlled by the feedback defined by (19) is also granted, provided that some left-invariant metric on Remark 1: Equations (18) and (19) are the coordinate-free generalizations of (5) and (6) derived for the three-dimensional chained system They hold whatever the system of coordinates chosen to represent the elements of as vectors in Remark 2: (Explicit Control Expression): The explicit calculation of the feedback control defined by (19) will often require the preliminary choice of a system of coordinates Once this choice is made, if one keeps the same letters to denote the vf involved in (19) and their (local) representations in, then (19) is equivalent to (21) the invertible matrix defined by (11),, and, denoting the local representation in of the point Equation (21) is the coordinate-dependent generalization of (6) Remark 3: (Output Linearization and Associated Zero Dynamics): Since the choice of in the control expression is free, one can interpret this vf as a free control vector and (21) as a feedback control which, in view of (20), almost linearizes the equations of the original system A clear advantage of this type of linearization over exact feedback linearization is that it is free of singularities Another one is the decoupled form of the resulting linear system and, as a matter of fact, its extreme simplicity This interpretation is conceptually attractive, but it is important to realize that it does not tell everything about the approach For instance, it leaves in the shade the dynamics associated the extra set of variables, while these hidden dynamics may have their importance in the overall evaluation of the control performance For example, if the convergence of to zero results in large sustained values of then highly oscillatory motion of will take place This may not be desirable for a certain number of applications On the other hand, oscillatory motion may also correspond to a requirement of the application Just consider the case of a car having to perform a small lateral motion in a given time period This can only be done via one or several maneuvers whose number and amplitude are clearly related to the frequency of the steering-wheel angular velocity Typically, the smaller the amplitude the larger the frequency will need to be Remark 4: (About the Convergence of to Zero): Proposition 1 indicates how any vf which asymptotically stabilizes the identity element of induces a feedback law for System (1) which asymptotically stabilizes the set For instance, given a system of coordinates on, exponential stability is obtained by choosing, any Hurwitz stable matrix Note that uniform asymptotic boundedness of all control variables is automatically ensured As already mentioned, a complementary analysis of the zero dynamics associated is usually necessary to deduce other properties of the closed-loop system The equations of this zero dynamics are just obtained by setting in the control expression (21) In the specific case when the perturbation is identically equal to zero, or when it does not depend upon while tends to zero, these equations simplify to and one deduces that all control variables converge to zero This implies in particular, that tends to zero Note that this does not necessarily mean that the state itself converges to some point However, it is not difficult to prove that such is the case when is identically equal to zero,

6 MORIN AND SAMSON: PRACTICAL STABILIZATION OF DRIFTLESS SYSTEMS ON LIE GROUPS 1501 or when tends exponentially to zero, and converges exponentially to zero Now, let us point out that may also tend to zero when the perturbation does not vanish In view of (19), this requires the existence of elements such that The possibility of having the extended state converge when does not vanish will be illustrated along the unicycle example treated further in this paper B Trajectory Stabilization We show how Proposition 1 directly applies to the problem of practical stabilization of a trajectory on a Lie group Let denote an arbitrary smooth trajectory in For any basis of, there exist smooth functions such that Furthermore, if are left-invariant vf in and are solutions to the differential equations then the following identity holds: One deduces the following error system associated the trajectory stabilization problem: representing the tracking error and (22) (23) Since this system has the same form as (1), Proposition 1 applies to it and provides control laws which ensure asymptotic stabilization of the set for the error system (22) It is in this sense that practical stabilization of the trajectory is achieved In particular, if denotes a left invariant Riemannian distance on, then the practical stabilization of implies that, for any solution of the controlled system, asymptotically tends to zero VI CASE OF HOMOGENEOUS SYSTEMS In this section, we show how the results of the previous sections apply to driftless controllable systems homogeneous vf The study of such systems is motivated by several reasons One of them is that any controllable smooth driftless control system on can be approximated by a controllable system homogeneous vf [42], [16] While this approximation is local in general, there are also physical systems which admit an homogeneous representation in a large domain The modeling by chained systems of the kinematic equations of several nonholonomic wheeled mobile robots is a well-known example The main tool used to apply the results of the previous sections to homogeneous systems is the so-called lifting theorem [38] which specifies how homogeneous systems can be viewed as systems on Lie groups This explains in part the importance given to Lie groups in the formal Lie-algebraic literature [19], [43] In this literature, free Lie algebras and free systems [18] are usually considered They correspond to the framework for the original lifting theorem where nilpotent vf are lifted to a free Lie group While this is well justified from a theoretical standpoint and the sake of generality, it can be interesting, for practical purposes and computational efficiency, to lift the vf associated a specific control system under consideration to the smallest possible Lie group, ie, the embedding Lie group the smallest dimension This possibility, investigated in [14], will be used here A Lifting of an Homogeneous System to a System on a Lie Group Prior to stating, in the form of a proposition, a version of the lifting theorem adapted to our present objectives, let us recall a few basic definitions and properties about homogeneity (for more details, we refer the reader to [16]) Given and a weight vector,adilation on is a map from to defined by A function is homogeneous of degree respect to the family of dilations, or more concisely -homogeneous of degree, if A smooth vf on is - homogeneous of degree if, for any, the function is -homogeneous of degree The possibility of lifting a set of homogeneous vf to a Lie group, dimension equal to the dimension of the Lie algebra generated by the vf, is summarized in the following proposition Proposition 2: [38], [14]: Let denote smooth vf on, independent 1 over -homogeneous of degree respectively, and which satisfy the LARC at the origin Let denote the dimension, over, of Then, there exist i) a lifting of on of the following form: ii) a smooth mapping ; such that 1) satisfy the LARC at the origin; 2) are -homogeneous of degree, respectively, for some ; 1 That is, ( X = 0 2 ;i = 1; ;m) =) ( = 0; 8i)

7 1502 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER ), endowed the composition law is a Lie group, and are left-invariant vf for this composition law For a constructive proof of this proposition, the interested reader is referred to [33] Remark 5: The combination of Theorem 1 and Proposition 2 implies the results stated in Theorem 1 of [34] and, in particular, the existence of transverse functions for any smooth controllable driftless system It also yields complementary results One of them is the specification of a smaller number of variables on which transverse functions depend More precisely, this number is equal to the difference between the dimension of the Lie algebra generated by the vf of a controllable homogeneous approximation of the system and the number of independent control vf Another important complementary result is the expressions (9) (10) of such a transverse function This expression is much more concise and explicit than the construction proposed in the proof of Theorem 1 in [34] Finally, let us mention that no general control expression was given in [34] In this respect the Lie group structure is essential The next section explains how to use Proposition 2 in conjunction Proposition 1 to achieve practical stabilization of a controllable homogeneous driftless system B Application to Practical Stabilization Consider a -dimensional controllable homogeneous driftless system perturbed by an additive vf (24) The control vf are assumed to be independent over so that all assumptions in Proposition 2 are satisfied Let be the vf evoked in this proposition A dynamic extension of system (24) is then the -dimensional system (25) Recall that the origin of is practically stabilized in the sense that the set is asymptotically stable for the extended system (25), after closing the loop the feedback control and choosing adequately This implies that the set denoting the first components of the vector-valued function is (globally) attractive for the closed-loop solutions of (24) Then, by using classical norm inequalities one easily verifies that, in order to further ensure stability of this set, it suffices to choose the initial condition equal to denoting the remaining components of the vector-valued function two in- VII EXAMPLES A Chained Systems The (perturbed) chained system of dimension puts is defined by (26), and some additive drift term The unperturbed chained system corresponding to the case where is a particular homogeneous system, and it is well known that is of dimension, and is spanned by the vf (27) and the th canonical vector on According to Proposition 2, and are left-invariant wrt a group operation on One easily verifies that this group operation is given by the matrix whose only nonzero entries are, for More explicitly, the components of are defined by (28) (29) According to Proposition 2, the corresponding unperturbed system (obtained by setting ) is left invariant on the Lie group endowed the group operation defined in the proposition Therefore, Proposition 1 applies directly to system (25), yielding a feedback control which ensures (global) practical stabilization of the origin, ie, the identity element, of for this system The expression of the control law depends on a transverse function associated the vf, and Theorem 1 provides an example of such a transverse function From there, it remains to specify conditions under which the obtained controller is also a (global) practical stabilizer for the initial system (24) Let us now proceed the computation of transverse functions for this system and complemented the vf defined by (27) form a graded basis of in the sense of Definition 1, and the associated weight-vector is given by Therefore, if we use (10) to calculate each, we deduce from (27) that From the definition of,, and (27), it is not difficult to obtain

8 MORIN AND SAMSON: PRACTICAL STABILIZATION OF DRIFTLESS SYSTEMS ON LIE GROUPS 1503 The expression of can then be calculated from (29) Then, for any transverse function, and any Hurwitz stable matrix, the choice in the control expression (21) yields the feedback defined, as before, by (11) and, denoting the first component of and respectively This control is a practical stabilizer of (26) This was the main result of [32] up to a small difference due to inverting the roles played by and, knowing that both choices and can be made B Systems on With Two Control Inputs Recall that is the set of rotation matrices in Endowed the classical product of matrices, it is a three-dimensional Lie group Its Lie algebra consists of the set of (3 3) skew-symmetric real matrices and is classically denoted as Moreover, the exponential of an element of coincides the classical matrix exponential of this element Consider the following underactuated control system on this group: (30) where denotes, as in the above example, the th canonical vector in, and is the operator associated the vector product, ie, Since, and Using the fact that, for any rotation matrix and any vector, we deduce from (33) that and the identity matrix A direct calculation yields Since the transversality condition is equivalent to the third component of being different from zero, the proof follows For the determination of a control expression one can use (19) This yields or, equivalently This control is a practical stabilizer of the identity matrix provided that the vf asymptotic stabilizes the identity matrix A well known possible choice is, and the rotation vector associated the rotation matrix, ie, the vector smallest Euclidean norm (corresponding to the angle of rotation) such that The domain in which this vf is differentiable and exponentially stabilizes the identity matrix is minus the set of rotation matrices angles of rotation equal to Using this expression of in the previous equality yields form a basis of, the Lie algebra generated by and is isomorphic to This basis is a graded basis in the sense of Definition 1, We calculate a transverse function by (9) (10), ie, denoting the rotation vector associated the matrix, ie, such that A practical stabilizer is thus given by or, equivalently after simple calculations by (31) Note that is the matrix of rotation angle about the axis unitary vector A direct calculation yields (32) where and denote the and functions, respectively Lemma 4: The function given by (32) is a transverse function for any Proof: From (31) (33) The domain of attraction in contains the set of rotation matrices angles of rotation smaller than C Unicycle Kinematic equations of the unicycle are (34), and For stabilization purposes, this system is often transformed, via a diffeomorphic change of coordinates and new control variables, to the three dimensional chained system two inputs The solution described in Section VII-A then applies directly However, the preliminary transformation into a

9 1504 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003 chained system is not necessary, and the control design can be performed easily by using natural system coordinates This is possible because, as this is well known, the above equations define a left-invariant control system on the Lie group The group operation is given by (35) the rotation matrix in of angle The vf, and, form a graded basis of, A transverse function can be obtained by using (9) (10), or the more general expression (9) (12), ie, After simple calculations, using (35) and the expressions of the vf and Fig 1 Unicycle s and reference vehicle s (x; y) trajectories (36) The transversality condition is equivalent to the fact that the matrix We next report simulation results about the tracking of a reference vehicle by the unicycle The posture of the reference vehicle is given by and the control is determined so as to stabilize to zero, is invertible for any Using the expressions of,, and (36), one obtains so that the transversality condition is satisfied for any Another transverse function is obtained by using the control vf of the homogeneous approximation of system (34) which, in the present case, is the chained system of dimension three (up to a reordering of the state variables) Application of (9) (12) then yields (37) One can verify that the transversality condition is satisfied this function for and From (35) representing the tracking error Such a control is given by (21), of course, replaced by The chosen transverse function is given by (37) and The chosen stabilizing vf is defined by The expression of can be obtained either by application of (23), or by direct computation of By setting one obtains All terms in the feedback expression (21) are now defined For the simulation, the control inputs, and for the reference vehicle, on the time interval, are given by A practical stabilizer of is then given by (21) after choosing a vf which asymptotically stabilizes the origin of One can take, for example,, any Hurwitz stable matrix, in order to achieve exponential stability Note that for and, the reference trajectory is not feasible by the unicycle (since ) Fig 1 shows the trajectories of the unicycle and the reference vehicle in the plane, the dashed arrows indicating the orientation of the reference vehicle Fig 2 shows the time evolution of the components of the tracking vector and the unicycle s control inputs and We remark that the unicycle does not oscillate when the reference vehicle follows portions of feasible trajectories,

10 MORIN AND SAMSON: PRACTICAL STABILIZATION OF DRIFTLESS SYSTEMS ON LIE GROUPS 1505 Then, there exist analytic functions such that, for any Furthermore, if and denote analytic functions such that and, then is an analytic function and Proof: Let denote the structural constants associated, ie, From Definition 1, one deduces that From this fact, one obtains by induction on Therefore (38) Fig 2 Tracking errors and unicycle s control inputs as in the case of a straight line, when, and an arc of circle, when The simulation further shows that, as mentioned in Section I, oscillations are not systematic when the reference trajectory is not feasible This is illustrated by the portion of trajectory corresponding to Sustained oscillations only occur during the first phase corresponding to, when practical tracking of the reference vehicle the precision specified by requires them APPENDIX A PROOFS OF LEMMAS 1 3 The following technical claims are used in the proofs of Lemmas 1 and 2 Claim 1: Let and denote two time-dependent left-invariant vf on, and solutions of and respectively Then is a solution of The proof follows by classical calculus on Lie groups Claim 2: Let denote a graded basis of the Lie algebra of a Lie group Let and (39) It follows from (38) that each is an analytic function of and Furthermore, if and are analytic functions of and, then is analytic and it follows, by considering the term of lowest order in (39), that Note that the equality is uniform wrt because is periodic wrt

11 1506 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003 Proof of Lemma 1: In order to lighten the notation, we denote by the derivative of wrt We will also use for for, and for, and denote by and the functions defined by (40) Proof of Lemma 2: From Claim 1, and relations (9) and (13), one deduces that From the fact that and (10) (45) With this notation, In order to calculate we will use the following classical result ( [15, p 105], for example): (41) (46) the operator defined by where and are defined according to (40) By application of Claim 2, for any Since, one obtains, by application of (41) for some analytic functions Moreover, if and are analytic functions then, By applying this property recursively, one deduces from (46) that (47) for some analytic functions which depend on, and are such that (48) (recall the notation ) From (47) In view of (40), and since aforementioned equality yields (by Definition 1), the so that, by (45), (15) is satisfied (49) (42) where By application of Claim 2 (50) for some analytic functions Lemma 1 follows by using (43) (44) in (42) such that (43) (44) Lemma 2 follows from this decomposition Indeed, from Lemma 1 and (48),, and in (50) are This proves the first relation of (16) Concerning the case where and vanishes at because of (48) This accounts for the sum in the right-hand side of equality (16) (up to higher-order terms) Now, because of (14), and too

12 MORIN AND SAMSON: PRACTICAL STABILIZATION OF DRIFTLESS SYSTEMS ON LIE GROUPS 1507 For the last case, accounts for the sum in the right-hand side of equality (16) and, asin the previous case From Lemma 1, accounts for the term Proof of Lemma 3: The notation is used in this proof The lemma is a direct consequence of the following property which will be proved by induction: Therefore, from (55), (58), and (59) Then, (51) for follows by choosing small enough so as to ensure that the function defined by (51) is different from zero Thus, we have proved by induction the existence of numbers ( ) such that choosing [see (52)] Let us first prove (51) for From Lemma 2, Since and, (51) follows and Let us now prove that if (51) is satisfied for ( ), then it is also satisfied for Let (52) In these equalities, denotes the vector involved in (51), whereas is a scalar design parameter whose value will be specified below By (52), Therefore, one easily obtains from Lemma 2 the following equalities: By definition where denotes the cofactor of From (53) (53) (54) (55) (56) From the induction hypothesis (51), and the fact that, by a recursive application of (52), (57) By assuming that are small enough so that, one deduces from (54) and (57) that From (54) and (56), for any (58) (59) yields, provided that is itself chosen small enough This is precisely the result of Lemma 3 REFERENCES [1] A Astolfi, Discontinuous control of nonholonomic systems, Syst Control Letters, vol 27, pp 37 45, 1996 [2] M K Bennani and P Rouchon, Robust stabilization of flat and chained systems, in Proc Euro Control Conf, 1995, pp [3] A M Bloch and S Drakunov, Stabilization and tracking in the nonholonomic integrator via sliding modes, Syst Control Lett, vol 29, pp 91 99, 1996 [4] A M Bloch, M Reyhanoglu, and N H McClamroch, Control and stabilization of nonholonomic dynamic systems, IEEE Trans Automat Contr, vol 37, pp , 1992 [5] R W Brockett, Asymptotic stability and feedback stabilization, in Differential Geometric Control Theory, R S Millman, R W Brockett, and H J Sussmann, Eds Boston, MA: Birkaüser, 1983 [6] F Bullo, N H Leonard, and A D Lewis, Controllability and motion algorithms for underactuated lagrangian systems on Lie groups, IEEE Trans Automat Contr, vol 45, pp , 2000 [7] C Canudas de Wit, B Siciliano, and G Bastin, Eds, Theory of Robot Control New York: Springer-Verlag, 1996 [8] C Canudas de Wit and O J Sørdalen, Exponential stabilization of mobile robots nonholonomic constraints, IEEE Trans Automat Contr, vol 37, pp , Nov 1992 [9] W L Chow, Uber systeme von linearen partiellen differential-gleichungen erster ordnung, Math Ann, vol 117, pp , 1939 [10] J-M Coron, Global asymptotic stabilization for controllable systems out drift, Math Control, Sig, Syst, vol 5, pp , 1992 [11] D M Dawson, J Hu, and P Vedagharba, An adaptive control for a class of induction motor systems, in Proc IEEE Conf Decision Control, 1995, pp [12] W E Dixon, D M Dawson, E Zergeroglu, and F Zhang, Robust tracking and regulation control for mobile robots, Int J Robust Nonlinear Control, vol 10, pp , 2000 [13] M Fliess, J Lévine, P Martin, and P Rouchon, Flatness and defect of nonlinear systems: Introductory theory and examples, Int J Control, vol 61, pp , 1995 [14] G B Folland, On the Rothschild-Stein lifting theorem, Commun Partial Differential Equat, vol 2, pp , 1977 [15] S Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces New York: Academic, 1978 [16] H Hermes, Nilpotent and high-order approximations of vector field systems, SIAM Rev, vol 33, pp , 1991 [17] Y Kanayama, A Nilipour, and C Lelm, A locomotion control method for autonomous vehicles, in Proc IEEE ICRA, 1988, pp [18] M Kawski, Nonlinear control and combinatorics of words, in Nonlinear Feedback and Optimal Control, B Jakubczyk and W Respondek, Eds New York: Marcel Dekker, 1998 [19] M Kawski and H J Sussmann, Noncommutative power series and formal Lie-algebraic techniques in nonlinear control theory, in Operators, Systems, and Linear Algebra, D Pratzel-Wolters, U Helmke, and E Zerz, Eds New York: Teubner, 1997, pp

13 1508 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL 48, NO 9, SEPTEMBER 2003 [20] G Lafferriere and H J Sussmann, A differential geometric approach to motion planning, in Nonholonomic Motion Planning, Z Li and J F Canny, Eds Norwell, MA: Kluwer, 1993 [21] N E Leonard and P S Krishnaprasad, Motion control of drift-free, left-invariant systems on Lie groups, IEEE Trans Automat Contr, vol 40, pp , 1995 [22] W Lin and P Radom, Recursive design of discontinuous controllers for uncertain driftless systems in power chained form, IEEE Trans Automat Contr, vol 45, pp , 2000 [23] W Liu, An approximation algorithm for nonholonomic systems, SIAM J Control Optim, vol 35, pp , 1997 [24] D A Lizárraga, Obstructions to the existence of universal stabilizers for smooth control systems, Math Control, Sign, Syst, 2003, to be published [25] D A Lizárraga, P Morin, and C Samson, Non-robustness of continuous homogeneous stabilizers for affine control systems, in Proc IEEE Conf Decision Control, 1999, pp [26] M Maini, P Morin, J-B Pomet, and C Samson, On the robust stabilization of chained systems by continuous feedback, in Proc IEEE Conf Decision Control, 1999, pp [27] P Martin and P Rouchon, Feedback linearization of driftless systems, Math Control, Sig, Syst, vol 7, pp , 1994 [28] R T M Closkey and R M Murray, Exponential stabilization of driftless nonlinear control systems using homogeneous feedback, IEEE Trans Automat Contr, vol 42, pp , 1997 [29] P Morin, J-B Pomet, and C Samson, Design of homogeneous timevarying stabilizing control laws for driftless controllable systems via oscillatory approximation of Lie brackets in closed-loop, SIAM J Conttol Optim, vol 38, pp 22 49, 1999 [30] P Morin and C Samson, Exponential stabilization of nonlinear driftless systems robustness to unmodeled dynamics, Control Optim Calculus Var, vol 4, pp 1 36, 1999 [31], Control of nonlinear chained systems From the Routh-Hurwitz stability criterion to time-varying exponential stabilizers, IEEE Trans Automat Contr, vol 45, pp , 2000 [32], Practical stabilization of a class of nonlinear systems Application to chain systems and mobile robots, in Proc IEEE Conf Decision Control, 2000, pp [33], Practical stabilization of driftless systems on lie groups, INRIA, Tech Rep 4294, 2001 [34], A characterization of the Lie algebra rank condition by transverse periodic functions, SIAM J Control Optim, vol 40, pp , 2002 [35] R M Murray and S S Sastry, Nonholonomic motion planning: Steering using sinusoids, IEEE Trans Automat Contr, vol 38, pp , 1993 [36] J-B Pomet, Explicit design of time-varying stabilizing control laws for a class of controllable systems out drift, Syst Control Lett, vol 18, pp , 1992 [37] J-B Pomet and C Samson, Exponential stabilization of nonholonomic systems in power form, IFAC Symp Robust Control Design, pp , 1994 [38] L P Rothschild and E M Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math, vol 137, pp , 1976 [39] C Samson, Velocity and torque feedback control of a nonholonomic cart, presented at the Int Workshop Adaptative Nonlinear Control: Issues in Robotics, 1990 [40], Control of chained systems Application to path following and time-varying point-stabilization, IEEE Trans Automat Contr, vol 40, pp 64 77, Jan 1995 [41] O J Sørdalen and O Egeland, Exponential stabilization of nonholonomic chained systems, IEEE Trans Automat Contr, vol 40, pp 35 49, Jan 1995 [42] G Stefani, Polynomial approximations to control systems and local controllability, in Proc IEEE Conf Decision Control, 1985, pp [43] H J Sussmann, Lie brackets and local controllability: A sufficient condition for scalar-input systems, SIAM J Control Optim, vol 21, pp , 1983 [44] H J Sussmann and W Liu, Limits of highly oscillatory controls and approximation of general paths by admissible trajectories, in Proc IEEE Conf Decision Control, 1991, pp [45] A R Teel, R M Murray, and G Walsh, Nonholonomic control systems: From steering to stabilization sinusoids, Int J Control, vol 62, pp , 1995 Pascal Morin was born in Avranches, France, in 1969 He received the Maîtrise degree from Université Paris-Dauphine, Paris, France, in 1990, and the Diplôme d Ingénieur and PhD degrees from Ecole des Mines de Paris, Paris, France, in 1992 and 1996, respectively He spent one year as a Postdoctoral Fellow in the Control and Dynamical Systems Department at the California Institute of Technology, Pasadena Since 1997, he has been Chargé de Recherche at the Institut National de Recherche en Automatique et Informatique (INRIA), Sophia, France His research interests include control of nonlinear systems and its applications to mechanical systems Claude Samson was born in Buenos Aires, Argentina, in 1955 After graduating from the Ecole Supérieure d Electricité, Paris, France, in 1977, he received the Docteur-Ingénieur and Docteur d Etat degrees from the University of Rennes, Rennes, Francem in 1980 and 1983, respectively He joined the Institut National de Recherche en Automatique et Informatique (INRIA) in 1981, where he is currently Directeur de Recherche and the head of the research group Icare whose activities are centered on robotics and related control issues His research interests are in control theory and its applications to the control of mechanical systems He is the coauthor, M Leborgne and B Espiau, of the book Robot Control The Task-Function Approach (Oxford, UK: Oxford University Press, 1991)